Foundations of Computer Science: Honors
Homework 2B
Problem 5 Show that the function f : N→N has the listed properties:
1. f(x) = 2x (one-to-one but not onto)
2. f(x) = x + 1 (one-to-one but not onto) .
3. f(x) = if x is odd then x - 1, else x + 1 (bijective)
Problem 6 Show that the product (a + bi)(c + di) of two complex numbers can be evaluated using just three real-number multiplications. You may use a few extra additions
Problem 7 Given a fixed function f : A → A. An element a ∈ A is called a fixed point of f if f(a) = a. Find the set of fixed points for each of the following functions. 1. f : A → A where f(x) = x
Problem 8 Let f(x) = x2 and g(x,y) = x+y. Find compositions that use the functions f and g for each of the following expressions.
1. (x + y)2
2. x2 + y2
3. (x + y + z)2
4. x2 + y2 + z2
